Proof theory is an area of logic that studies proof as formal mathematical objects. For questions asking how to write proofs or for checking an informal proof, please use the proof-writing or proof-strategy tags instead.

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Is it decidable whether there are finitely or infinitely many positive integers n such that the (2^n)+1th last digit of 3^(2^n) in base 2 is 1.

It seems so obvious that there are infinitely many such n because using a probabilistic arguement, it seems like there's no chance of there only being finitely many of them. Yet, it seems impossible ...
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Next step to reach the contradiction?

This is a problem from Discrete Mathematics and its Applications Here are my notes and my current work so far for this problem. I started with an assumption that what i am trying to prove is ...
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What to use for r in proof by contradiction?

This is a problem from Discrete Mathematics and its applications To this proof, I am trying to use proof by contradiction. Here is how the book described the process of proof by contradiction. I ...
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Are proofs for many-sorted first order logic shorter than single sorted first order logic?

I understand that the expressive power of first order logic with one sort is the same as any many sorted first order logic, and that higher order logic with general semantics is the same as a many ...
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What is the fastest growing primitive-recursive-function?

Fast growing functions tend to be not primitive-recursive. So I wonder if there is a limit how fast a function can grow, if it is known that it is primitive recursive. What is the fastest growing ...
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57 views

What is the slowest growing function that cannot be proven to be total by PA?

I asked the question if PA can prove any function growing faster than $f_{\epsilon_0}(n)$ to be total. The answer was no. What about the converse : Can prove PA every function growing slower than ...
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54 views

Inference rule for Non-Empty Domains

I am currently experimenting with logic frameworks. I am basically using something along dependent types as in "Proof-assistants using Dependent Type Systems" by Henk Barendregt and Herman Geuvers. ...
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81 views

Is there any unreachable result?

I hope that this question is reasonable and make sense because I am not sure. Every theorem's proof is consisting of finite logical steps. Can a proof of the theorem require infinitely many ...
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Is the following derivation of Gödel's Paradox generalizable to an intuitionistic one using provability?

I will follow Goedel's original proof of Goedel's paradox located on pg. 264 of Hao Wang's A Logical Journey: From Gödel to Philosophy. With this in sight, and to also avoid confusion, I will also ...
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Can PA prove very fast growing functions to be total?

The Goodstein-sequence is a total function, but PA cannot prove this. Is this true for any other function with growth rate at least $f_{\epsilon_0}$ or are there functions growing at least as fast ...
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66 views

Proof in sequent calculus without cut

I met an exercise in Gaisi Takeuti, Proof Theory [Exercise 2.7, page 14]. How to construct a cut-free proof of$\ \forall xA(x)\rightarrow B\vdash \exists x(A(x)\rightarrow B)$, where A(a) and B are ...
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133 views

2 Questions regarding Relative Consistency Proofs

First Question: Let IC be the statement "There is an inaccessible cardinal." I have read that one cannot prove (in ZFC) the relative consistency of ZFC + IC w.r.t. ZFC. i.e. $ Con(ZFC) \rightarrow ...
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fake proof of $\forall a. \forall b. a = b \to 1 = 0$

I saw a less formal version of this fake proof that claimed to prove $2=1$ but because it assumed $a=b$ from the start I knew why it was wrong. It does seem however that the proof can be used to prove ...
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1answer
68 views

Beginning Haskell - cannot understand proof

I've just started reading "Thinking Functionally with Haskell" by Richard Bird In the preface he states : And after stating the proof he also states the proof will be used throughout the book. ...
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30 views

What does this proposition mean?

$∀x ∈ P(\Bbb{N}), x \notin \{\} \Rightarrow ∃y ∈ x, ∀z ∈ x \ | \ y < z$ Where $P(x)$ is the power set. I'm interpreting it as "in all subsets of the natural numbers, there exists a value smaller ...
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100 views

How to check homeomorphic embedding relation programmatically?

This is a follow up to this question and Deedlit's answer. I'm looking for a precise definition of the "hem?" (tree A homeomorphically embeddable in tree B?) relation, preferably in terms of a ...
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Help with understanding this proof in discrete mathematics?

This is the question and solution: Q: Prove that for any integer $a$, $2a + 1$ and $4a^2$ + 1 are relatively prime. A: Since $4a^2 + 1 = (2a − 1)(2a + 1) + 2$, any common divisor of $2a + 1$ and ...
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Proof by induction of this formula? [duplicate]

$2^0+2^1+2^2+...+2^n$ for $n ∈ \mathbb{N}$ U ${0}$. I made a conjecture that this is $2^{n+1} - 1$. Now I have to prove it by induction. I tested the base case where it's equal to zero, and it ...
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Z / 6Z being a set of well dedfined equivalence classes, and a congruent to b(mod 6)

why is this = [0],[1],[2],[3],[4],[5],[6] and how would I define f Z/6Z - Z/6Z by f([a]) = ([2a]). I have the proof but I don't understand it. Proof: Assume [a1] = [a2] in Z/6Z. then a1 congruent to ...
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1answer
47 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
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29 views

Proving Theorem: subspace of polynomials of degree two or less?

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less? I know I need to show that $a+b+c=0$ ...
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1answer
26 views

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$.

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$. Prove that $f$ is well-defined, surjective and injective I don't really have a problem with figuring out if it's ...
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Proof by Induction Divisibility.

$6^n-5n+4$ is divisible by 5 for all positive integers $n$. $n >=1$ Prove By Induction My attempt is as follows: $n=1$ $6^1-5(1) +4$ $=5$, Therefore 5 is divisible by 5 so $n=1$ is true ...
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Uncountable reals in the theory

The Question I'm looking for a possibility to somehow proof the "essence" of Cantor's diagonal argument within a recursive first-order theory which is satisfied by the reals (better: within a theory ...
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How far can-I rewrite in lambda functions?

I am quite new with the lambda calculus. I am experimenting lambda-calculus proofs through the coq proof assistant, but the question I have is not related to coq (I guess). However, I'm going to use ...
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53 views

Euclid's proof for infinitely many prime numbers

Prove that there are infinitely many primes congruent to 3mod4 using euclid's proof for infinitely many prime number. I guess I don't really know where to start because I don't understand euclid's ...
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1answer
58 views

Comparing Statements and predicates using Truth Tables

Consider the four statements: $∃x$ $∀y$ $p(x, y)$ $∃y$ $∀x$ $p(x, y)$ $∀x$ $∃y$ $p(x, y)$ $∀y$ $∃x$ $p(x, y)$ which we call S1, S2, S3 and S4 respectively. Does there exist a predicate p such ...
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Proving $f\colon S \to S$; $f(x) = 1/x$ is bijective

Hey I'm trying to figure out this proof. I don't know if anyone could help but I would really appreciate it! Let $S = \mathbb{R} \setminus \{0\}$. Prove that the function $f\colon S \to S$; $f(x) = ...
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Model-theoretic question about language of field theory.

Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite ...
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1answer
59 views

Proof Strategy, Power sets, and Injections

Let $S$ and $T$ be sets and define the function $$f:\mathcal P(S) \times \mathcal P (T)\to \mathcal P(S \cup T)$$ by $f(A,B) = A \cup B$ for all $A \subseteq S$ and all $B \subseteq T$. Prove that ...
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How to find the shortest proof of a provable theorem?

Roughly speaking, there are some fundamental theorems in mathematics which have several proofs (e.g. Fundamental Theorem of Algebra), some short and some long. It is always an interesting question ...
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how to know when a particular proof is appropriate for the given problem?

The main trouble I am currently having in math is knowing when the use cases are appropriate in a proof. I see many videos where they seem to choose a strategy like proof by contrapositive or proof by ...
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Undecidability and truth

Are there undecidable problems for which a single truth exists? For example, the question about parallels is not decidable from Euclid axioms. But multiple answers are valid and give different kinds ...
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Proving $\forall x (A\to B) \to(A \to \forall x B):x\notin \mbox{free}(A)$ in a Hilbert system where it is not an axiom

I have no idea whether this question is way too specific or whether something similar has already been asked (we still need to work out a way to search for formulas I guess). Anyways here I go: I ...
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Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
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115 views

Is the parallelogram law a theorem or an axiom?

I'm learning about inner product spaces and I am able to prove it within an inner product space. Is this a theorem or an axiom in euclidean geometry?(note: not the geometry of Descartes)
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Trouble understanding algebra in induction proof

I'm on hour 20 of studying for the discrete math midterm tomorrow, and I've got to be honest I'm a little panicked. In particular I'm having trouble with induction proofs, not because I don't ...
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43 views

How can Goodstein's theorem be expressed in PA

I understand Goodstein's Theorem and its proof. I'm trying to understand the proof of why Goodstein's Theorem cannot be proved in PA. However, it's not immediately clear to me that Goodstein's Theorem ...
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Proof by contradiction how to show is properly

For every $x \in \left[\pi/2,\pi\right]\,,\ \sin\left(x\right) − \cos\left(x\right) \geq 1$. I have drawn the graph and can clearly see that A is true however how do I prove it correctly.
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If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

Let $n\in\mathbb{N}$. So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$. Help?
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Euclidean algorithm to provde gcd's and multiples

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b). I was going to try setting it up, by literally doing: nb = rna + k and so forth, but something tells me this ...
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Proving the dimension of the intersection of 2 subspaces

Assume that $U$ and $W$ are distinct subspaces $( U ≠ W )$ of a four-dimensional vector space $V$ and $\dim(U) = \dim(W) = 3$. Prove that $\dim ( U ∩ W ) = 2$.
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How Do I Show that Condensed Derivable Rules of Inference Yield the Same Formula as Using Condendensed Detachment Multiple Times?

If we look at condensed detachment of two formulas $\alpha$ and $\beta$, we can see that D$\alpha$.$\beta$, where $\alpha$ has form C$\alpha$$_a$$\alpha$$_b$ is equivalent to using the rule ...
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268 views

Why wouldn't someone accept Gentzen's consistency proof?

Reading the consistency section of the Peano Axioms wikipedia page, I came across this sentence: The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, ...
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Proof of law of reflection using Fermat's principle : are we really proving the law of reflection?

Before you skip reading this, let me tell you that this isn't a "how to derive the law of reflection using Fermat's principle" question. Also, I asked it on MSE instead of the physics site because ...
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Is there a proof for what I describe as the “recursive process of mathematical induction for testing divisibility”.

I was working on my homework for Discrete Math, and we were asked to "Prove: $6 | n^{3}+5n$,where $n\in \mathbb{N}$" my solution varied significantly from how I have seen it done by others. I noticed ...
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1answer
62 views

Question about the incompleteness proof (Theorem V)

Question in short: Where do I find a complete proof of Theorem V from Gödels incompleteness proof? If it does not exists, can someone provide it? Question in detail: I am trying to understand ...
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Proving Properties in Ordered Fields

Refer to Definition 1.3, which states, an ordered field is a field F that is ordered set with the following additional properties: ...
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Using logical Properties to prove a tautology

So I have to prove this as a tautology. I've been stuck on this forever and am not sure where to go. I experimented and got this far, and looking for some pointers on where to take it next. (p → q) ...
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128 views

Birkhoff's completeness theorem

I have two simple questions. A) Does Birkhoff's completeness theorem follow directly from Gödel's completeness theorem? B) Is Birkhoff's completeness theorem constructive in the following sense: ...